Abstract: This is joint work with Yair Glasner.

A sufficiently rich action of a group G on a topological space X can be used to study properties of G, and in particular the representations of G as a permutation group and the collection of maximal subgroups of G. We gave a criterion which determines when G admits a faithful primitive permutation action (i.e. one which do not preserve any non-trivial equivalence relation). This criterion can be applied in many different settings such as linear groups, hyperbolic groups, mapping class groups, etc. and its consequences generalize theorems of Platonov, Wehrfritz, Ivanov and Kapovich.

In this talk I will explain our method for the simple case of groups acting on trees, and deduce a proof for a conjecture of Higman and Neumann.